Oracle Based Active Set Algorithm for Scalable Elastic Net Subspace Clustering

• Published in: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) pp. 3928 - 3937
• Main Authors: Chong You, Chun-Guang Li, Robinson, Daniel P., Vidal, Rene
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.06.2016
• Subjects: Clustering algorithms; Computer vision; Data models; Dictionaries; Geometry; Motion segmentation; Optimization
• ISSN: 1063-6919
• DOI: 10.1109/CVPR.2016.426

State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with ℓ 1 , ℓ 2 or nuclear norms. ℓ 1 regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. ℓ 2 and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed ℓ 1 , ℓ 2 and nuclear norm regularizations offer a balance between the subspace-preserving and connectedness properties, but this comes at the cost of increased computational complexity. This paper studies the geometry of the elastic net regularizer (a mixture of the ℓ 1 and ℓ 2 norms) and uses it to derive a provably correct and scalable active set method for finding the optimal coefficients. Our geometric analysis also provides a theoretical justification and a geometric interpretation for the balance between the connectedness (due to ℓ 2 regularization) and subspace-preserving (due to ℓ 1 regularization) properties for elastic net subspace clustering. Our experiments show that the proposed active set method not only achieves state-of-the-art clustering performance, but also efficiently handles large-scale datasets.